# A “class group” obstruction for the equation $C{y}^{d}=F\left(x,z\right)$

Denis Simon[1]

• [1] LMNO - UMR 6139 Université de Caen – France Campus II – Boulevard Mal Juin BP 5186 – 14032 Caen Cedex
• Volume: 20, Issue: 3, page 811-828
• ISSN: 1246-7405

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## Abstract

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In this paper, we study equations of the form $C{y}^{d}=F\left(x,z\right)$, where $F\in ℤ\left[x,z\right]$ is a binary form, homogeneous of degree $n$, which is supposed to be primitive and irreducible, and $d$ is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result can be used to prove that these equations have no proper solution. Numerous examples are given to illustrate this result. In a second part, we make a link between this condition and the properties of the different in the considered number field.

## How to cite

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Simon, Denis. "A “class group” obstruction for the equation $Cy^d=F(x,z)$." Journal de Théorie des Nombres de Bordeaux 20.3 (2008): 811-828. <http://eudml.org/doc/10862>.

@article{Simon2008,
abstract = {In this paper, we study equations of the form $Cy^d = F(x,z)$, where $F\in \mathbb\{Z\}[x,z]$ is a binary form, homogeneous of degree $n$, which is supposed to be primitive and irreducible, and $d$ is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result can be used to prove that these equations have no proper solution. Numerous examples are given to illustrate this result. In a second part, we make a link between this condition and the properties of the different in the considered number field.},
affiliation = {LMNO - UMR 6139 Université de Caen – France Campus II – Boulevard Mal Juin BP 5186 – 14032 Caen Cedex},
author = {Simon, Denis},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {811-828},
publisher = {Université Bordeaux 1},
title = {A “class group” obstruction for the equation $Cy^d=F(x,z)$},
url = {http://eudml.org/doc/10862},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Simon, Denis
TI - A “class group” obstruction for the equation $Cy^d=F(x,z)$
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 3
SP - 811
EP - 828
AB - In this paper, we study equations of the form $Cy^d = F(x,z)$, where $F\in \mathbb{Z}[x,z]$ is a binary form, homogeneous of degree $n$, which is supposed to be primitive and irreducible, and $d$ is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result can be used to prove that these equations have no proper solution. Numerous examples are given to illustrate this result. In a second part, we make a link between this condition and the properties of the different in the considered number field.
LA - eng
UR - http://eudml.org/doc/10862
ER -

## References

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1. B.J. Birch, H.P.F. Swinnerton-Dyer, Notes on elliptic curves. J. Reine Angew. Math. 212 (1963), 7–25. Zbl0118.27601MR146143
2. J.W.S. Cassels, Rational Quadratic Forms. L.M.S. Monographs, Academic Press, 1978. Zbl0395.10029MR522835
3. J.E. Cremona, Algorithms for Modular Elliptic Curves. Cambridge University Press, 1997, Second Edition. Zbl0872.14041MR1628193
4. J.E. Cremona, Elliptic Curve Data. http://modular.math.washington.edu/cremona/INDEX.html Zbl01.0253.01
5. H. Cohen, Number Theory. Vol I : Tools and Diophantine Equations. GTM 239, Springer Verlag, 2007. Zbl1119.11001MR2312337
6. H. Darmon, A. Granville, On the equations ${z}^{m}=F\left(x,y\right)$ and $A{x}^{p}+B{y}^{q}=C{z}^{r}$. Bull. London Math. Soc 27 (1995), 513–543. Zbl0838.11023MR1348707
7. I. Delcorso, R. Dvornicich, D. Simon, The decomposition of primes in nonmaximal orders. Acta Arithmetica 120 (2005), 231–244. Zbl1163.11342MR2188842
8. F. G. Frobenius, Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1896), 689–703; Gesammelte Abhandlungen II, 719–733. Zbl27.0091.04
9. E. Hecke, Vorlesungen über die Theorie der algebraischen Zahlen, 2te Unveränderte Auflage (1954), Leipzig, Akademische Verlagsgesellschaft (1923). Zbl0057.27301MR66417
10. S. Lang, Algebraic number theory, GTM 110, second edition. New York, Springer–Verlag, 1994. Zbl0811.11001MR1282723
11. MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/magma/
12. J. R. Merriman, S. Siksek, N. P. Smart, Explicit 4-descent on an elliptic curve. Acta Arithmetica 77 (1996), 385–404. Zbl0873.11036MR1414518
13. pari/gp, The Pari group (K. Belabas, H. Cohen,...). http://pari.math.u-bordeaux.fr/
14. I. Reiner, Maximal Orders. L.M.S. Monographs, Academic Press, 1975. Zbl0305.16001MR1972204
15. D. Simon, La classe invariante d’une forme binaire. Comptes Rendus Mathématiques 336, Issue 1 , (2003) 7–10. Zbl1038.11073MR1968893

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